Find the limits in a), b), and c) below for the function $$ f(x)=\frac{4 x}{x-7} $$. Use $$ -\infty $$ and $$ \infty $$ when appropriate. a) Select the correct choice below and fill in any answer boxes in your choice. A. $$ \lim _{x \rightarrow 7^{-}} f(x)=\square $$ (Simplify your answer.) B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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UpStudy AI · Accepted Answer

Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 7^{-}}\left(\frac{4x}{x-7}\right)$$
- step1: Rewrite the expression:
  $$\frac{\lim _{x\rightarrow 7^{-}}\left(4x\right)}{\lim _{x\rightarrow 7^{-}}\left(x-7\right)}$$
- step2: Calculate:
  $$\frac{28}{\lim _{x\rightarrow 7^{-}}\left(x-7\right)}$$
- step3: Calculate:
  $$-\infty$$
Calculate the limit $$ \lim_{x \rightarrow 7^{+}} \frac{4x}{x-7} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 7^{+}}\left(\frac{4x}{x-7}\right)$$
- step1: Rewrite the expression:
  $$\frac{\lim _{x\rightarrow 7^{+}}\left(4x\right)}{\lim _{x\rightarrow 7^{+}}\left(x-7\right)}$$
- step2: Calculate:
  $$\frac{28}{\lim _{x\rightarrow 7^{+}}\left(x-7\right)}$$
- step3: Calculate:
  $$+\infty$$
To find the limits for the function $$ f(x) = \frac{4x}{x-7} $$:

### a) 
1. **Left-hand limit as $$ x $$ approaches 7**:
   - We calculate $$ \lim_{x \rightarrow 7^{-}} f(x) $$:
   - As $$ x $$ approaches 7 from the left, $$ x - 7 $$ approaches 0 and is negative, making the denominator negative. Therefore, the function approaches $$ -\infty $$.
   - Thus, $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$.

2. **Right-hand limit as $$ x $$ approaches 7**:
   - We calculate $$ \lim_{x \rightarrow 7^{+}} f(x) $$:
   - As $$ x $$ approaches 7 from the right, $$ x - 7 $$ approaches 0 and is positive, making the denominator positive. Therefore, the function approaches $$ +\infty $$.
   - Thus, $$ \lim_{x \rightarrow 7^{+}} f(x) = +\infty $$.

### Conclusion for a)
- The correct choice is:
  - A. $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$

### b)
- The limit does not exist at $$ x = 7 $$ because the left-hand limit and right-hand limit are not equal (one approaches $$ -\infty $$ and the other $$ +\infty $$).

### Final Answers:
- A. $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$
- B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.
a) $$ \lim_{x \rightarrow 7^{-}} f(x) = -\infty $$ b) The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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